Solving a logarithmic expression without a calculator

Question

How do I find the value of this logarithmic expression without using a calculator? I'm trying to relearn algebra, but this problem has me scratching my head, and my Google tutorial searches are failing me.

$2^{\log_2 10}$

Answer 1

If the base of the logarithm is $10$:

Using the identity $\log_a a=1$ we have the following:

$$2^{\log_{10} {10}}=2$$

If the base of the logarithm is $2$:

Using the identity $a^{\log_a x}=x$ we have the following:

$$2^{\log_{2} {10}}=10$$

Answer 2

If you meant $2^{\log_2{10}}$, then the answer is simply $10$:

$\log_2{10}$ means, a value $x$ such that $2^x=10$.

So if you take this $x$ and calculate $2^x$, then you will obviously get $10$.